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We show that this assumption mismatches with the multi-dimensional structure of model features, provably inducing feature splitting through two distinct mechanisms. Geometrically, reconstructing a feature of intrinsic dimension $d_i \\ge 2$ to error $\\varepsilon$ with single-direction decoders forces a number of atoms that is exponential in $d_i$. From an end-to-end optimization perspective, this splitting is not merely possible but actively preferred. We prove that there exists a continuous path from the true $d_i$-dimensional basis to a strictly lower risk of the $\\ell_1$-regularized SAE objective, whose descent directions drive any trained dictionary into that exponential regime. 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We prove that there exists a continuous path from the true $d_i$-dimensional basis to a strictly lower risk of the $\\ell_1$-regularized SAE objective, whose descent directions drive any trained dictionary into that exponential regime. 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